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- 2. What does it mean that preferences are complete? Now would the real-life implica- tions of this assumption look like?For the case of two goods, give an example of a utility function U(x1, x2) that represents the preferences of a consumer who regards the two goods as perfect complements. Next, take the transformation f(U) = U³ of the your example utility function and explain if this newly gener ated function represents the original preferences. Further, provide clear arguments supporting or rejecting the claim that "f(U(x1, x2)) must be strongly increasing in (x1, x2)."2. Ping receives a weekly allowance to purchase soda (x) and juice (y) at school. His preferences over these two goods can be represented by the utility function U(x,y) =0.5x+ 5lny where x represents the number of cans of soda and y represents the number of juice boxes. a) Given his preferences find his Marshallian demand functions for soda (x) and juice (y). b) Suppose that the price of a can of soda is $1.5 and that Ping has $30 to spend on soda and juice. Write Ping's demand curve for juice (y). Illustrate his demand curve. c) Suppose that the price of a box of juice is $1 (the price of soda and income remain $1.5 and $30, resp.). Use your demand functions to find his best bundle. In an indifference curve diagram illustrate his best bundle at these prices. For the remainder of the question assume that Ping's allowance (income) rises to $36 per week and the prices of the two goods are unchanged at Px= $1.5 and Py= $1. d) Use your demand functions to find his new best bundle. e)…
- 2 3 1. The consumer has Cobb-Douglas preferences for two goods x and y: u(x, y) = 10x¾yš Py> 0 and m > 0 is given. a) Set up the Expenditure Minimization Problem. b) Calculate MU and MUy. Ꮖ c) State the condition that explain why Cobb-Douglas preferences will guarantee an interior solu- tion. c) Give the two optimality conditions for x and y. i) ii) Px > 0, e) Solve for the compensated demand functions for good x and y.Let the following table represents the total utility of a given consumer, in the cardinal utility approach A) Calculate the MUx and MUy and fill the table in the 4th and 5th rows. B) If the two products (X&Y) are free goods how many of X and Y should the cons consumer take to maximize utility? C) What is the maximum utility of X and Y if they are free?. D) Assuming the consumer has any amount of money (enough budget) how many of X and Y should the consumer buy, to maximize utility? E) What is the total utility of X and Y? F) Let now price of X is 4 birr per unit and price of Y is 2 birr per unit and budget of the consumer for consumption of X and Y is 20 birr. Given budget constraint how many of X and Y should the consumer buy to maximize utility? G) What are the total utility of X and Ya. Is the “more is better” assumption satisfied for both goods? Explain.b. What type of function is this? Explain.c. Provide expressions for the marginal utilities of x and y and does the marginal utility of x increase, remain constant, or diminish as the consumer buysmore units of x? Explain. Does the marginal utility of y increase, remain constant, or diminish as the consumer buysmore units of y? Explain.d. Provide an expression for the marginal rate of substitution of x for y.e. Is MRSx,y diminishing, constant, or increasing as the consumer substitutes more x for y alongan indifference curve? Explain.f. On a graph with x on the horizontal axis and y on the vertical axis, draw the indifferencecurves for U1 and U2 where U2 has a higher value than U1 instructions: answer d,e, and f only
- a. Is the “more is better” assumption satisfied for both goods? Explain.b. What type of function is this? Explain.c. Provide expressions for the marginal utilities of x and y and does the marginal utility of x increase, remain constant, or diminish as the consumer buysmore units of x? Explain. Does the marginal utility of y increase, remain constant, or diminish as the consumer buysmore units of y? Explain.d. Provide an expression for the marginal rate of substitution of x for y.e. Is MRSx,y diminishing, constant, or increasing as the consumer substitutes more x for y alongan indifference curve? Explain.f. On a graph with x on the horizontal axis and y on the vertical axis, draw the indifferencecurves for U1 and U2 where U2 has a higher value than U14. The CES utility function is given by x°+y° U (x, y) (i). Show that the first-order conditions for a constrained utility maximum with this function require individuals to choose goods in the proportion = Pa 8–1 Py (ii). Show that the result in part (i) implies that individuals will allocate their funds equally between x and y for the case 8 = 0. (iii). How does the ratio depend on the PyY value of 8? Explain your results intuitively. (iv). Derive the indirect utility and expenditure functions for this case and check your results by describing the homogeneity properties of the functions you calculated.2. Suppose John's utility function is UX,Y) = 4/XY , where X is consumption of beer and Y is consumption of pizza. For this utility function, ( Calculus Kahuna say: MUx = 2Y/X and MUy = 2\X/Y ) a) Suppose Y=3. Calculate John's utility for X=4,9,16, and 25. For a given level of Y, does good X display diminishing marginal utility?
- A consumer has the following utility function: Ulx, y) = xy -y, *21 where x and y represents the quantities consumed of goods X and Y. y 20 What will be the substitution and income effects for X and Yassuming that the consumer attempts to maintain the same level of utility achieved before price of Y increased (that is, when price of Y was $1)? SEx= +0.5 IEx = -0.5 SE, = -0.25 IE- = -0.25 SEx= +0.293 IE = -0.293 SEy = -0.414 IE, = +0.414 SEr= +0.25 IE SE, = -0.75 IE, = -0.75 = -0.25 SEx= +0.414 IEx = -0.414 SEy = -0.293 IE, = -0.207 Income = $3 Px= $1, Py= $22. In this chapter, we showed an example in which the consumer has preferences for consump- tion with the perfect complements property. Suppose, alternatively, that leisure and con- sumption goods are perfect substitutes. In this case, an indifference curve is described by the equation i = al+ bC, where a and b are positive constants, and u is the level of utility. That is, a given indifference curve has a particular value for u, indifference curves having higher values for u. (a) Show what the consumer's indifference cur- with higher ves look like when consumption and leisure are perfect substitutes, and determine graphically and algebraically what con- sumption bundle the consumer chooses. Show that the consumption bundle the con- sumer chooses depends on the relationship between a/b and w, and explain why. (b) Do you think it likely that any consumer would treat consumption goods and leisure as perfect substitutes?a. Determine the demand functions of x and y in the case of a Cobb-Douglas type utility function, in the following cases: α=0.40;β=0.60 Graph the demand functions of the two goods (price as a function of quantity) assuming the individual's income is $500 - Determine what is the quantity demanded of x and y, if the price of good x is USD 1, the price of good y is USD 4, and income is USD 500 - Now, explain what happens to the quantity demanded if the prices of the goods are doubles holding income constant.